Integrand size = 11, antiderivative size = 98 \[ \int \cos \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )}{\sqrt {c}} \]
1/2*cos(a-1/4*b^2/c)*FresnelC(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))*2^(1 /2)*Pi^(1/2)/c^(1/2)-1/2*FresnelS(1/2*(2*c*x+b)/c^(1/2)*2^(1/2)/Pi^(1/2))* sin(a-1/4*b^2/c)*2^(1/2)*Pi^(1/2)/c^(1/2)
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.87 \[ \int \cos \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{2}} \left (\cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )-\operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right ) \sin \left (a-\frac {b^2}{4 c}\right )\right )}{\sqrt {c}} \]
(Sqrt[Pi/2]*(Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])] - FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)]))/Sqrt[c]
Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3929, 3832, 3833}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cos \left (a+b x+c x^2\right ) \, dx\) |
\(\Big \downarrow \) 3929 |
\(\displaystyle \cos \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx-\sin \left (a-\frac {b^2}{4 c}\right ) \int \sin \left (\frac {(b+2 c x)^2}{4 c}\right )dx\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \cos \left (a-\frac {b^2}{4 c}\right ) \int \cos \left (\frac {(b+2 c x)^2}{4 c}\right )dx-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {\sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelC}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}-\frac {\sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b^2}{4 c}\right ) \operatorname {FresnelS}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {2 \pi }}\right )}{\sqrt {c}}\) |
(Sqrt[Pi/2]*Cos[a - b^2/(4*c)]*FresnelC[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]) /Sqrt[c] - (Sqrt[Pi/2]*FresnelS[(b + 2*c*x)/(Sqrt[c]*Sqrt[2*Pi])]*Sin[a - b^2/(4*c)])/Sqrt[c]
3.1.3.3.1 Defintions of rubi rules used
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Cos[(b^2 - 4* a*c)/(4*c)] Int[Cos[(b + 2*c*x)^2/(4*c)], x], x] + Simp[Sin[(b^2 - 4*a*c) /(4*c)] Int[Sin[(b + 2*c*x)^2/(4*c)], x], x] /; FreeQ[{a, b, c}, x] && Ne Q[b^2 - 4*a*c, 0]
Time = 0.36 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {C}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )+\sin \left (\frac {\frac {b^{2}}{4}-a c}{c}\right ) \operatorname {S}\left (\frac {\sqrt {2}\, \left (c x +\frac {b}{2}\right )}{\sqrt {\pi }\, \sqrt {c}}\right )\right )}{2 \sqrt {c}}\) | \(81\) |
risch | \(\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {i \left (4 a c -b^{2}\right )}{4 c}} \operatorname {erf}\left (\sqrt {i c}\, x +\frac {i b}{2 \sqrt {i c}}\right )}{4 \sqrt {i c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {i \left (4 a c -b^{2}\right )}{4 c}} \operatorname {erf}\left (-\sqrt {-i c}\, x +\frac {i b}{2 \sqrt {-i c}}\right )}{4 \sqrt {-i c}}\) | \(99\) |
1/2*2^(1/2)*Pi^(1/2)/c^(1/2)*(cos((1/4*b^2-a*c)/c)*FresnelC(2^(1/2)/Pi^(1/ 2)/c^(1/2)*(c*x+1/2*b))+sin((1/4*b^2-a*c)/c)*FresnelS(2^(1/2)/Pi^(1/2)/c^( 1/2)*(c*x+1/2*b)))
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05 \[ \int \cos \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {2} \pi \sqrt {\frac {c}{\pi }} \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) - \sqrt {2} \pi \sqrt {\frac {c}{\pi }} \operatorname {S}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {\frac {c}{\pi }}}{2 \, c}\right ) \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )}{2 \, c} \]
1/2*(sqrt(2)*pi*sqrt(c/pi)*cos(-1/4*(b^2 - 4*a*c)/c)*fresnel_cos(1/2*sqrt( 2)*(2*c*x + b)*sqrt(c/pi)/c) - sqrt(2)*pi*sqrt(c/pi)*fresnel_sin(1/2*sqrt( 2)*(2*c*x + b)*sqrt(c/pi)/c)*sin(-1/4*(b^2 - 4*a*c)/c))/c
Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \cos \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {2} \sqrt {\pi } \left (- \sin {\left (a - \frac {b^{2}}{4 c} \right )} S\left (\frac {\sqrt {2} \left (b + 2 c x\right )}{2 \sqrt {\pi } \sqrt {c}}\right ) + \cos {\left (a - \frac {b^{2}}{4 c} \right )} C\left (\frac {\sqrt {2} \left (b + 2 c x\right )}{2 \sqrt {\pi } \sqrt {c}}\right )\right ) \sqrt {\frac {1}{c}}}{2} \]
sqrt(2)*sqrt(pi)*(-sin(a - b**2/(4*c))*fresnels(sqrt(2)*(b + 2*c*x)/(2*sqr t(pi)*sqrt(c))) + cos(a - b**2/(4*c))*fresnelc(sqrt(2)*(b + 2*c*x)/(2*sqrt (pi)*sqrt(c))))*sqrt(1/c)/2
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int \cos \left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + \left (i + 1\right ) \, \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + i \, b}{2 \, \sqrt {i \, c}}\right ) + {\left (\left (i + 1\right ) \, \cos \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + \left (i - 1\right ) \, \sin \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \operatorname {erf}\left (\frac {2 i \, c x + i \, b}{2 \, \sqrt {-i \, c}}\right )\right )}}{8 \, \sqrt {c}} \]
-1/8*sqrt(2)*sqrt(pi)*(((I - 1)*cos(-1/4*(b^2 - 4*a*c)/c) + (I + 1)*sin(-1 /4*(b^2 - 4*a*c)/c))*erf(1/2*(2*I*c*x + I*b)/sqrt(I*c)) + ((I + 1)*cos(-1/ 4*(b^2 - 4*a*c)/c) + (I - 1)*sin(-1/4*(b^2 - 4*a*c)/c))*erf(1/2*(2*I*c*x + I*b)/sqrt(-I*c)))/sqrt(c)
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.38 \[ \int \cos \left (a+b x+c x^2\right ) \, dx=\frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} i \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {i \, b^{2} - 4 i \, a c}{4 \, c}\right )}}{4 \, {\left (\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} - \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{4} i \, \sqrt {2} {\left (2 \, x + \frac {b}{c}\right )} {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}\right ) e^{\left (-\frac {-i \, b^{2} + 4 i \, a c}{4 \, c}\right )}}{4 \, {\left (-\frac {i \, c}{{\left | c \right |}} + 1\right )} \sqrt {{\left | c \right |}}} \]
1/4*I*sqrt(2)*sqrt(pi)*erf(-1/4*I*sqrt(2)*(2*x + b/c)*(I*c/abs(c) + 1)*sqr t(abs(c)))*e^(-1/4*(I*b^2 - 4*I*a*c)/c)/((I*c/abs(c) + 1)*sqrt(abs(c))) - 1/4*I*sqrt(2)*sqrt(pi)*erf(1/4*I*sqrt(2)*(2*x + b/c)*(-I*c/abs(c) + 1)*sqr t(abs(c)))*e^(-1/4*(-I*b^2 + 4*I*a*c)/c)/((-I*c/abs(c) + 1)*sqrt(abs(c)))
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int \cos \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {2}\,\sqrt {\pi }\,\cos \left (\frac {4\,a\,c-b^2}{4\,c}\right )\,\mathrm {C}\left (\frac {\sqrt {2}\,\left (\frac {b}{2}+c\,x\right )\,\sqrt {\frac {1}{c}}}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{c}}}{2}-\frac {\sqrt {2}\,\sqrt {\pi }\,\sin \left (\frac {4\,a\,c-b^2}{4\,c}\right )\,\mathrm {S}\left (\frac {\sqrt {2}\,\left (\frac {b}{2}+c\,x\right )\,\sqrt {\frac {1}{c}}}{\sqrt {\pi }}\right )\,\sqrt {\frac {1}{c}}}{2} \]